We consider a Euclidean vector space $(E, (-\mid-))$. Given $a \in E$ and $x \in E$, we denote by $a \otimes x$ the map from $E$ to itself defined by:
$$\forall z \in E, (a \otimes x)(z) = (a \mid z) \cdot x$$
We fix $x \in E \backslash \{0\}$. Show that the map $a \in E \mapsto a \otimes x$ is linear and constitutes a bijection from $E$ onto $\{u \in \mathcal{L}(E) : \operatorname{Im} u \subset \operatorname{Vect}(x)\}$.